JEE Main · 2020 · 08 Jan 2020 · Shift 2 · Parabola

Let a line

y = m x

(m > 0)

intersect the parabola,

y^{2} = x

at a point P, other than the origin. Let the tangent to it at P meet the

x

-axis at the point Q. If area

(△ O P Q) = 4

sq. units, then

m

is equal to

Answer: 0.5

Solution

The line

y = m x

meets

y^{2} = x

where

(m x)^{2} = x

, that is at

P = (\frac{1}{m ^{2}}, \frac{1}{m})

. Writing

P = (t^{2}, t)

with

t = \frac{1}{m}

, the tangent to

y^{2} = x

P

t y = \frac{x + t ^{2}}{2}

, which meets the

x

-axis at

Q = (- t^{2}, 0) = (- \frac{1}{m ^{2}}, 0)

. The triangle

O P Q

has area

\frac{1}{2} ∣ x_{P} y_{Q} - x_{Q} y_{P} ∣ = \frac{1}{2} \cdot \frac{1}{m ^{3}}

, and setting this equal to

4

gives

m^{3} = \frac{1}{8}

, so

m = \frac{1}{2}

JEE Main · 2020 · 08 Jan 2020 · Shift 2 · Numerical answer

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